Klasse 6ad

1. Exam: 12.09.2025

Curve Sketching and Optimization Problems

Relevant Exercises (Rhyn): Chapter 3, Section a) Exercises 18, 19, 20, 21, Section b) Exercises 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, Section c) Exercises 40, 41, 42, 43, 44, Section d) Exercises 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71; Chapter 5, Section d) Exercises 44, 45, 46, 47, 48, 49, 50, 51, 51, 52, 53, 54, 56, 57, 58.

Learning Goals:

• Be able to determine stationary points using the first derivative.

• Be able to apply the concept of the second derivative geometrically to describe curvature behavior.

• Be able to calculate inflection points using the second derivative.

• Be able to determine the type of a stationary point using the second derivative (maximum, minimum, saddle point).

• Be able to determine, with the help of the second derivative, whether a function is curved to the left (convex) or curved to the right (concave) in a given interval.

• Be able to set up the function equation of polynomial functions from information given in a text, using differential calculus.

• Be able to carry out complete curve sketching for the following functions: polynomial functions, rational functions, root functions, trigonometric functions.

• Be able to calculate vertical (poles), horizontal, and oblique (slant) asymptotes of rational functions.

• Be able to apply polynomial division to determine oblique (slant) asymptotes of rational functions and to find zeros of higher-order polynomials.

• Be able to recognize optimization problems and solve them using algebraic and geometric methods.

• Be able to identify objectives and objective functions from information given in a text.

• Be able to reduce objective functions of two variables with boundary conditions (constraints) to functions of a single variable.

• Be able to optimize objective functions using the first derivative.

3. Exam: 27.03.2026

Stochastics

Relevant Exercises (DMK Book): Chapter 1, Exercises 1–16, 18, 20–25, 29; Chapter 2, Exercises 2–23; Chapter 3, Exercises 1–18; Chapter 5, Exercises 1–18; Chapter 6, Exercises 1–9

Learning Goals:

• apply basic concepts of set theory (set, subset, union, intersection, complement) to events and model simple random experiments using a sample space

• describe events as subsets of the sample space and calculate probabilities of events

• solve combinatorial problems using factorials and binomial coefficients and distinguish between permutations, variations, and combinations

• apply the binomial theorem and interpret binomial coefficients

• represent multi-stage random experiments using tree diagrams and determine probabilities using the product rule and the sum rule

• define random variables and determine their probability distributions

• calculate and interpret the expected value of a discrete random variable

• calculate variance and standard deviation and interpret them as measures of dispersion of random variables

• recognize and model Bernoulli trials (Bernoulli chains)

• calculate probabilities using the binomial distribution and determine the expected value and standard deviation of the binomial distribution.